
with \(k^\mu /m_V={\cal O}(\lambda )\)  or \({\cal O}(\lambda ^2)\) . While an on-shell vector boson satisfies the constraint \(v\cdot \varepsilon _V=0\)  trivially, the time-like mode \(v\cdot V\)  might still propagate in EFT loop diagrams, but it cannot do so as a hard degree of freedom, since hard quantum fluctuations are – by definition – already integrated out. In order to separate the relevant scales in a consistent way, one needs to decompose the resonance into an on-shell component and soft fluctuations of small virtualities, using a construction analogous to HQET [1]}, [2]}, [3]}.
